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Properties

Label 2-392-56.51-c2-0-59
Degree $2$
Conductor $392$
Sign $0.386 + 0.922i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (0.5 + 0.866i)3-s + (−1.99 + 3.46i)4-s + (−4.5 − 2.59i)5-s + (−0.999 + 1.73i)6-s − 7.99·8-s + (4 − 6.92i)9-s − 10.3i·10-s + (−8.5 − 14.7i)11-s − 3.99·12-s + 13.8i·13-s − 5.19i·15-s + (−8 − 13.8i)16-s + (−12.5 − 21.6i)17-s + 16·18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.166 + 0.288i)3-s + (−0.499 + 0.866i)4-s + (−0.900 − 0.519i)5-s + (−0.166 + 0.288i)6-s − 0.999·8-s + (0.444 − 0.769i)9-s − 1.03i·10-s + (−0.772 − 1.33i)11-s − 0.333·12-s + 1.06i·13-s − 0.346i·15-s + (−0.5 − 0.866i)16-s + (−0.735 − 1.27i)17-s + 0.888·18-s + (−0.184 + 0.319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.629107 - 0.418471i\)
\(L(\frac12)\) \(\approx\) \(0.629107 - 0.418471i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (4.5 + 2.59i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.5 + 14.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + (12.5 + 21.6i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (4.5 + 2.59i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 13.8iT - 841T^{2} \)
31 \( 1 + (-28.5 + 16.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (7.5 + 4.33i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 26T + 1.68e3T^{2} \)
43 \( 1 - 14T + 1.84e3T^{2} \)
47 \( 1 + (43.5 + 25.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (79.5 - 45.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (27.5 + 47.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-19.5 - 11.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (8.5 + 14.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (-59.5 - 103. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (64.5 + 37.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 110T + 6.88e3T^{2} \)
89 \( 1 + (-35.5 + 61.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 22T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.24509086023174588898716878903, −9.693242891035269000202620461642, −8.786299267600420450285894563314, −8.179386281007294493833791948818, −7.09556685196472592971858596132, −6.18907881980744969914125106165, −4.87787489538121890389904306521, −4.12485153570406238698165241736, −3.07372198726198901966868104219, −0.26782422019622277127815803120, 1.84791382442180117280096629163, 2.97123713603629619893186660059, 4.23493425879817195637086330745, 5.06606486702189315410521818035, 6.51742771387250535138479745994, 7.64379315621861629148135533605, 8.351307241013879604590193160189, 9.890795826965897415081309256873, 10.51139586354733376306350403441, 11.17766965070117784851339366073

Graph of the $Z$-function along the critical line